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Properties

Label 20.0.48316237149...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{15}\cdot 41^{19}$
Root discriminant $765.96$
Ramified primes $2, 5, 41$
Class number $23262013216$ (GRH)
Class group $[2, 4, 2907751652]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![787722016100000, 0, 857751578200000, 0, 312165040270000, 0, 47512954140000, 0, 3577377715000, 0, 143862932000, 0, 3199197200, 0, 39237000, 0, 256250, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 256250*x^16 + 39237000*x^14 + 3199197200*x^12 + 143862932000*x^10 + 3577377715000*x^8 + 47512954140000*x^6 + 312165040270000*x^4 + 857751578200000*x^2 + 787722016100000)
 
gp: K = bnfinit(x^20 + 820*x^18 + 256250*x^16 + 39237000*x^14 + 3199197200*x^12 + 143862932000*x^10 + 3577377715000*x^8 + 47512954140000*x^6 + 312165040270000*x^4 + 857751578200000*x^2 + 787722016100000, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 256250 x^{16} + 39237000 x^{14} + 3199197200 x^{12} + 143862932000 x^{10} + 3577377715000 x^{8} + 47512954140000 x^{6} + 312165040270000 x^{4} + 857751578200000 x^{2} + 787722016100000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4831623714907498679729543628551708670427136000000000000000=2^{55}\cdot 5^{15}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $765.96$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3280=2^{4}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3280}(1,·)$, $\chi_{3280}(961,·)$, $\chi_{3280}(1281,·)$, $\chi_{3280}(717,·)$, $\chi_{3280}(77,·)$, $\chi_{3280}(1837,·)$, $\chi_{3280}(1041,·)$, $\chi_{3280}(1813,·)$, $\chi_{3280}(409,·)$, $\chi_{3280}(2649,·)$, $\chi_{3280}(1437,·)$, $\chi_{3280}(1973,·)$, $\chi_{3280}(613,·)$, $\chi_{3280}(2409,·)$, $\chi_{3280}(237,·)$, $\chi_{3280}(1841,·)$, $\chi_{3280}(1333,·)$, $\chi_{3280}(2729,·)$, $\chi_{3280}(1849,·)$, $\chi_{3280}(213,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4}$, $\frac{1}{10} a^{5}$, $\frac{1}{10} a^{6}$, $\frac{1}{10} a^{7}$, $\frac{1}{300} a^{8} + \frac{1}{3}$, $\frac{1}{300} a^{9} + \frac{1}{3} a$, $\frac{1}{300} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{300} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3000} a^{12} + \frac{1}{30} a^{4}$, $\frac{1}{3000} a^{13} + \frac{1}{30} a^{5}$, $\frac{1}{3000} a^{14} + \frac{1}{30} a^{6}$, $\frac{1}{249000} a^{15} + \frac{1}{6225} a^{13} - \frac{1}{1660} a^{11} + \frac{37}{24900} a^{9} + \frac{23}{1245} a^{7} + \frac{2}{1245} a^{5} - \frac{11}{83} a^{3} + \frac{118}{249} a$, $\frac{1}{7470000} a^{16} + \frac{29}{249000} a^{14} - \frac{1}{49800} a^{12} + \frac{1}{6225} a^{10} - \frac{37}{74700} a^{8} - \frac{11}{1245} a^{6} - \frac{11}{2490} a^{4} - \frac{68}{249} a^{2} - \frac{1}{9}$, $\frac{1}{1247490000} a^{17} - \frac{23}{41583000} a^{15} + \frac{559}{5197875} a^{13} + \frac{87}{138610} a^{11} + \frac{104}{3118725} a^{9} - \frac{6149}{415830} a^{7} + \frac{8647}{207915} a^{5} - \frac{585}{13861} a^{3} + \frac{44506}{124749} a$, $\frac{1}{498200705358292688527962608259265557502332690000} a^{18} - \frac{6665442224355521766648823935341218119307}{99640141071658537705592521651853111500466538000} a^{16} + \frac{623628880968427335621019421426924307501137}{16606690178609756284265420275308851916744423000} a^{14} - \frac{2701800670229433792368528962189348345402447}{16606690178609756284265420275308851916744423000} a^{12} + \frac{536879500089369809108469710679896535367841}{1245501763395731721319906520648163893755831725} a^{10} + \frac{6442754229345946769289061841996848088803}{99640141071658537705592521651853111500466538} a^{8} + \frac{2478850165070442608433490120878428447921761}{83033450893048781421327101376544259583722115} a^{6} - \frac{1172524487270181448416468202058593280086237}{166066901786097562842654202753088519167444230} a^{4} - \frac{6980030148816171536935289846882944804954643}{49820070535829268852796260825926555750233269} a^{2} + \frac{1706598928187597118289811250403869875554}{3594262357393353210648312591149740693329}$, $\frac{1}{498200705358292688527962608259265557502332690000} a^{19} - \frac{45031234231948972649642556247953828403}{124550176339573172131990652064816389375583172500} a^{17} - \frac{2683515787704738121217641012140156915719}{8303345089304878142132710137654425958372211500} a^{15} + \frac{234539988206579760241550510937695312685927}{2767781696434959380710903379218141986124070500} a^{13} + \frac{6586432011738270451584544892789515897732679}{4982007053582926885279626082592655575023326900} a^{11} - \frac{2148045027649503360097611609730945540983067}{2491003526791463442639813041296327787511663450} a^{9} + \frac{1310470945874213918264922985606058370023431}{33213380357219512568530840550617703833488846} a^{7} + \frac{2659755399873868486646982130380004378057453}{55355633928699187614218067584362839722481410} a^{5} + \frac{4575523330203459035299035133663471524098092}{49820070535829268852796260825926555750233269} a^{3} + \frac{18302911730965425908936330925560393148165632}{49820070535829268852796260825926555750233269} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{4}\times C_{2907751652}$, which has order $23262013216$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13054936686.08412 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{410}) \), 4.0.17643776000.8, 5.5.2825761.1, 10.10.33523910081941606400000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
41Data not computed